Definition:Composition of Mappings/Definition 3

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Definition

Let $S_1$, $S_2$ and $S_3$ be sets.

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.


The composite of $f_1$ and $f_2$ is defined and denoted as:

$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \map {f_1} x = y \land \map {f_2} y = z}$


That is:

$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \tuple {x, y} \in f_1 \land \tuple {y, z} \in f_2}$


CompositeMapping.png


Commutative Diagram

The concept of composition of mappings can be illustrated by means of a commutative diagram.

This diagram illustrates the specific example of $f_2 \circ f_1$:

$\begin{xy}\xymatrix@+1em{ S_1 \ar[r]^*+{f_1} \ar@{-->}[rd]_*[l]+{f_2 \mathop \circ f_1} & S_2 \ar[d]^*+{f_2} \\ & S_3 }\end{xy}$


Warning

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that:

$\Dom {f_2} \ne \Cdm {f_1}$

where $\Dom {f_2}$ and $\Cdm {f_1}$ denote domain and codomain respectively.

Then the composite mapping $f_2 \circ f_1$ is not defined.


Compare with the definition of composition of relations in the context of the fact that a mapping is a special kind of relation.


Also known as

In the context of analysis, this is often found referred to as a function of a function, which (according to some sources) makes set theorists wince, as it is technically defined as a function on the codomain of a function.

Some sources call $f_2 \circ f_1$ the resultant of $f_1$ and $f_2$ or the product of $f_1$ and $f_2$.


Some authors write $f_2 \circ f_1$ as $f_2 f_1$.

Some use the notation $f_2 \cdot f_1$ or $f_2 . f_1$.

Some use the notation $f_2 \bigcirc f_1$.

Others, particularly in books having ties with computer science, write $f_1; f_2$ or $f_1 f_2$ (note the reversal of order), which is read as (apply) $f_1$, then $f_2$.


Also see

  • Results about composite mappings can be found here.


Sources


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